5.1.1. Potential flow streamlines versus smoke streaklines

The smoke streaklines corresponding to $\alpha = 0$ and $S_D=0$ serve as a base case to be compared with $|S_D|\gt 0$ cases, where $\psi (r,\theta )$ -contours drawn as red lines are overlaid on the streaklines in figure 6(a). Similarly, black potential flow streamlines are overlaid on white PIV streamlines in figure 6(b), which will be discussed further in § 5.2. The domain of the streakline image in figure 6(a) matches the streamwise PIV field of view, so a direct side-by-side qualitative comparison can be made. The streaklines and the potential flow streamlines are in fair qualitative agreement, especially near the cone vertex. The disparities between the streaklines and streamlines emerge downstream of the cone vertex, which may be attributed to the three dimensionality of the smoke visualisation experiment and difficulties visualising in the symmetry plane. Note that the free stream is uniform in the experiments, which is not the case in the potential flow solution; this also contributes to the disparity with the streaklines. The cone model vertex is rounded to a 2 mm radius; therefore, the potential flow streamlines in both plots of figure 6 are offset upstream of the cone to simulate a cone model with a sharp vertex.

Figure 6. $\psi (r,\theta )$ streamlines overlaid on streaklines and PIV streamlines at $y\approx 0$ corresponding to $\theta _c=10^\circ$ , $S_D=0$ and $\alpha =0$ . The cone height is $H/D=2.83$ .

5.2.1. Streakline behaviours near the surface

For the spinning cases, $S_D=\pm 2$ , at zero incidence, $\alpha = 0$ , the streakline patterns are nearly identical to the non-spinning case outside of the boundary layer; hence, the effects of rotation are confined within the boundary layer. Streaklines that come in close proximity with the boundary layer are ingested and wrapped around the cone surface in the direction of rotation. At angles of incidence $\alpha \leqslant 6^\circ$ , the flows remains attached to the leeward surface of the cone, where streaklines contacting the windward surface further reveal the effects of rotation.

Details corresponding to $S_D = 0 \text{ and } 2$ exhibit different behaviours from those corresponding to $S_D = -2$ . The streaklines near the surface have similar orientations and coherency for $S_D = 0 \text{ and } 2$ cases, whereas the streaklines corresponding to $S_D = -2$ are more blurry and faint. This is a result of counter-flow at the counter-rotating meridian of the cone (as defined in figure 1), where the direction of angular rotation is in opposition to the incoming free stream. The counter-flow acts by slowing flows near the surface and widening the smoke streaklines, which may be thought of as stream tubes. As the stream tubes are shrunk in the streamwise direction, they grow in width, giving the smoke streaklines their blurry appearance. This behaviour is mainly visible in $\alpha \gt \theta _c$ cases (rows 3–7 of figure 5).

In most cases, the streaklines embrace the $-y$ -meridian of the cone; however, since the streaklines form in two slightly separated planes, there are instances where streaklines follow the $+y$ -meridian. This effect is most noticeable in $S_D\lt 0$ snapshots where ingested streaklines in the rotating boundary layer follow the direction of rotation in the $+y$ -meridian and re-emerge from the leeward surface ( $\alpha = 6^\circ , 12^\circ , 24^\circ \text{ and }30^\circ$ corresponding to rows $2, 3, 5\text{ and }6$ of figure 5). The re-emerging streaklines form deep in the boundary layer and generate a cross-hatch pattern with the streaklines that form further away and follow the direction of the reference flow corresponding to $S_D = 0$ .

For the $S_D=2$ cases, the streaklines near the surface are increasingly curved in the direction of rotation as $\alpha$ increases. At $\alpha = 18^\circ \text{ and } 24^\circ$ , streaklines following the co-rotating meridian near the leeward surface are ejected in the direction of rotation; these streaklines exhibit sinusoidal patterns.

To measure the oscillation frequency of the trailing tip vortex, a stationary vertical line probe that intersects the vortex path is employed. By tracking the pixel intensity along the vertical ( $z$ ) direction over time, the $z$ -position of the vortex centre is determined at each frame. With known image acquisition rate, the periodic motion in the $z$ -direction is analysed to extract its frequency.

The frequency of these signatures match the angular rotation rate $\varOmega$ ; slight misalignment of the cone axis and the axis of the servomotor shaft of ${\sim} 1$ mm influences the trajectories of these streaklines. The influence of misalignment emanates from the vertex of the cone, as streaklines in contact with the cone vertex exhibit periodic waves.

5.2.2. Flow separation and vortical flows

At angles of incidence $\alpha \geqslant 12^\circ$ , flow separates near the leeward surface; it is likely that this transition occurs at $\alpha \approx \theta _c$ . The first case where flow separation is noticeable corresponds to $\alpha = 12^\circ$ (row 3 in figure 5), where a region devoid of streaklines forms close to the leeward surface. For the non-spinning case at $\alpha = 12^\circ$ , a helical pattern develops from the streaklines near the leeward surface. This helical pattern highlights the camera side ( $-y$ direction) of the expected trailing vortex system. Streaklines in regions above the helical pattern are curved downwards towards the surface, highlighting the induced down-wash effect of the trailing vortex system; in contrast with the behaviour corresponding to $\alpha \leqslant 6^\circ$ , where the streaklines near the leeward surface are approximately parallel to the surface. The separation angle parameter for $\theta _c=10^\circ$ is estimated to be $\varLambda \approx 0.9$ since the precise value of $\alpha _*$ is between $6^\circ$ and $12^\circ$ .

For $\alpha \gt 12^\circ$ , vortices in the separated flow regions become increasingly prevalent ( $\alpha = 18^\circ , 24^\circ , 30^\circ \text{ and }36^\circ$ corresponding to rows 4–7 in figure 5). For $S_D=0$ cases, a symmetric vortex pair aligns itself nearly parallel to the surface; it emerges near the cone vertex, grows monotonically in size in the $+x_*$ -direction and remains close to the surface along the entire length of the cone. At the highest case, $\alpha = 36^\circ$ , the interiors of the symmetric vortex are visible and composed of an indiscernible number of distinct helical structures; this artefact is further discussed in § 5.2.

The co-rotating meridian streaklines near the leeward surface are drawn into the separation region by the effects of rotation for flows corresponding to $S_D=2$ . For $\alpha \gt 24^\circ$ , they leave the camera’s field of view as they enter the counter-rotating meridian. In the highest angle of incidence case, $\alpha =36^\circ$ , a blurry streakline tube is visible in the $+y$ -meridian of the cone, a signature of a trailing vortex in the counter-rotating meridian.

Vortices in the counter-rotating meridian are increasingly visible in the $S_D=-2$ cases for $18^\circ \leqslant \alpha \leqslant 36^\circ$ . At the lowest case, $\alpha = 18^\circ$ , a vortex emerges near the vertex of the cone, grows in diameter in the $+x_*$ -direction and remains close to the surface of the cone as it curves in the direction of rotation. As $\alpha$ is increased past $18^\circ$ , it forms further from the surface in the $+z_*$ -direction. At $\alpha = 30^\circ$ , the vortex detaches from the surface roughly at $x_*/D = 2.0$ or $x/D \approx 1.9$ , and at $\alpha = 36^\circ$ , the vortex detaches roughly at $x_*/D = 1.1$ or $x/D = 1.0$ .

Small-scale wave patterns form on the detached sections of the vortices at $\alpha = 30^\circ \text{ and } 36^\circ$ . Furthermore, the helical patterns are coherent near the vertex of the cone, but their coherency diminish in detached sections of the vortices. Note that the vortex visualised in the $\alpha = 36^\circ$ for $S_D=-2$ has the same orientation as the blurred streakline tube visualised in the $S_D=2$ case (row 7 figure 5). The helical patterns visualised in $S_D=-2$ cases exhibit the same periodic waves attributed to misalignment between the central axis of the cone and the axis of the servomotor shaft.

5.2.3. Influence of $S_D$ on vortical flows

Figure 7. Smoke streakline images corresponding to $\theta _c=10^\circ$ , ${\textit{Re}}_D = 8.8 \times 10^3$ , ${\textit{Re}}_L = 2.5\times 10^4$ , $|S_D|=1, 1.5, 1.7, 2, 2.5\text{ and }3$ , and $\alpha = 36^\circ$ .

To explore the effects of varying $S_D$ , smoke streaks are captured at $\alpha = 36^\circ$ for $S_D = \pm [ 1, 1.5, 1.7, 2, 2.5, 3]$ and are shown in figure 7. The video sequences associated with each case up to $S_D = \pm 2.5$ show the same periodic wave behaviour seen in figure 5; in each case, the frequencies of the waves match the corresponding angular speed $\varOmega$ , leading to the conclusion that the waves are a result of misalignment of the axes of the cone and the motor. At the highest rotational speed parameter $S_D=\pm 3$ (row 6 of figure 7), however, the periodic wave is less noticeable.

Except for the case $\alpha = 36^\circ$ and $S_D=2$ in figure 5, helical patterns are visible in the remaining $S_D\gt 0$ cases in figure 7. They originate near the vertex of the cone and follow the direction of rotation towards the counter-rotating meridian, eventually leaving the field of view. The helical patterns highlight a vortex forming in the co-rotating meridian, which consistently forms underneath a sheet of streaklines following the direction of rotation. As the rotational speed ratio increases from 1 to 3, the co-rotating meridian vortex is visible up to a monotonically decreasing distance $x_*/D \approx 2.4 \rightarrow 0.9$ or $x/D \approx 2.1 \rightarrow 0.8$ . The helical pattern forming in the co-rotating meridian is visible in all $S_D\gt 0$ snapshots in figure 7 with $S_D=2$ as an exception; this is likely due to the smoke-wire being shifted too far from the plane of symmetry and hence, regions influenced by the trailing vortex systems. Glimpses of a counter-rotating meridian vortex are also visible in the $S_D\gt 0$ cases of figure 7. In snapshots of the $S_D = 1, 1.7\text{ and }2$ cases, the vortex is marked by a diffuse streakline tube that grows in thickness as the rotational speed parameter increases. In snapshots of the $S_D = 2.5 \text{ and } 3$ cases, exquisite details of the vortex are revealed; characterised by a blurry streakline tube marking its centreline and thinner streaklines wrapping around it helically.

Visualisations of the counter-rotating meridian vortex are shown in the $S_D\lt 0$ cases in the right column of figure 7; their orientations match their $S_D\gt 0$ counterparts in the left column. The vortices on the counter-rotating meridian align themselves parallel to the leeward surface of the cone until they detach from the surface. The detachment distance decreases from $x_*/D \approx 1.9 \rightarrow 0.76$ or $x/D \approx 1.7 \rightarrow 0.68$ as $S_D$ increases from $1$ to $3$ ; hence, an increase in $|S_D|$ expedites the detachment of vortices in the counter-rotating meridian. As observed for $\alpha = 36^\circ$ and $S_D=2$ in figure 5, the helical patterns lose their coherency in regions of the detached vortex over the full range of $S_D$ . Small-scale wave patterns are also present on the detached portions of the vortices.

By tracing the origins of helical patterns beneath the counter-rotating meridian vortex in $S_D\lt 0$ snapshots, they are observed to emerge at distances $x_*/D$ where the co-rotating meridian side vortex leaves the field of view in the corresponding $S_D\gt 0$ snapshots. Thus, the co-rotating meridian vortices maintain their integrity as they are pulled into the counter-rotating meridian. A unique snapshot revealing both co-rotating and counter-rotating meridian vortices corresponds to $S_D=-3$ , row 6 column 2 of figure 7. Both vortices are distinctly marked by separate sets of streaklines forming coherent helical patterns. Exquisite details of the counter-rotating meridian vortex (the set of helical patterns furthest from the cone) shows evidence of a vortex merging event, where the primary core of the vortex is wrapped by a secondary set of helical-shaped streaklines.

5.2.4. Oblique measurements

The topology of the trailing vortex systems are revealed by the oblique vorticity contours studied here, where the vortex systems exhibit distinct self-similar arrangements for the non-spinning and spinning cases. Schematics of the trailing vortices behind a non-spinning and spinning cone of half-angle $\theta _c = 10^\circ$ are shown in figures 8(a) and 8(b), respectively, where the definitions used to describe them are outlined. Figures 9, 10, 11 and 12 show $\omega _xD/U_\infty$ -contours with overlaid streamlines coloured in magenta computed from oblique plane PIV measurements corresponding to $\alpha = 18^\circ$ , $24^\circ$ , $30^\circ$ and $36^\circ$ , respectively. Vorticity contour plots corresponding to $\alpha$ near $\theta _c$ are omitted from the text due to faint signatures of the trailing vortex systems. Measurements are not made for $\alpha \ll \theta _c$ cases due to physical constraints of the camera and laser sheet alignment. Each figure is composed of measurements made in $yz$ -planes at $x/D \in [0, 2.38]$ in $0.34$ increments, where each row corresponds a discrete $x/D$ slice; and each column to a discrete $S_D$ .

Figure 8. Sketches of the trailing vortex systems behind non-spinning and spinning cones. (a) Primary, secondary and tertiary vortices of the symmetric vortex triads for $S_D=0$ cases. (b) Cyclonic and anti-cyclonic vortex triads with respect to the co-rotating and counter-rotating meridians for $|S_D|\gt 0$ cases.

Figure 9. $\omega _x D/U_\infty$ in $yz$ -planes corresponding to $\theta _c=10^\circ$ , $S_D\in [0,2]$ and $\alpha = 18^\circ$ .

Figure 10. $\omega _x D/U_\infty$ in $yz$ -planes corresponding to $\theta _c=10^\circ$ , $S_D\in [0,2]$ and $\alpha = 24^\circ$ .

Figure 11. $\omega _x D/U_\infty$ in $yz$ -planes corresponding to $\theta _c=10^\circ$ , $S_D\in [0,2]$ and $\alpha = 30^\circ$ .

Figure 12. $\omega _x D/U_\infty$ in $yz$ -planes corresponding to $\theta _c=10^\circ$ , $S_D\in [0,2,3]$ and $\alpha = 36^\circ$ .

5.2.5. Symmetric vortex systems

Based on observations from the oblique vorticity contours (figures 9–12) a symmetric vortex system emerges near the leeward surface of the cone for non-spinning cones at angles of incidence $\alpha \gtrsim \theta _c$ . Near the vertex of the cone, $x/D\approx 0$ , the vortex systems are mainly composed of a symmetric pair of primary vortices. The primary vortices quickly develop into vortex triads with increasing $x/D$ ; each triad consists of primary and secondary vortices with nearly circular cross-sections and a tertiary vortex with an elliptic cross-section. The primary and tertiary vortices share the same sign of $\omega _x$ , and the secondary forms in between them with opposite sign. The primary vortices form from detached shear layers, which only exist in cases were flow separation is observed, i.e. when $\alpha \geqslant \alpha _*$ . As the primary vortices form, they are pushed towards the surface by their induced flow fields, apparent from the streamlines overlaid the vorticity contours in figures 9–12. Secondary vortices form from viscous effects near the surface, which split the primary vortices; hence, the formation of a tertiary vortex. The schematic of the symmetric vortex system in figure 8(a) describes the topology of $\omega _x$ -contours corresponding to $S_D=0$ cases.

In flows corresponding to $S_D=0$ , the cross-sectional size of the vortices depicted by $\omega _x$ -contours grow as $\alpha$ increases. The vortex strengths associated to the symmetric vortex triads are computed, where the vortex strengths of the left-side ( $-y$ ) and right-side ( $+y$ ) symmetric vortex triads are defined as $\varGamma _{\textit{left}}$ and $\varGamma _{\textit{right}}$ , respectively.

The vortex strength of each triad (2.5) is computed over an area enclosing the full triad, including only vorticity of the same sign as the primary and tertiary vortices; opposite-signed secondary vortices are excluded via a sign-based filter.

The non-dimensional values $|\varGamma _{\textit{left}}|/DU_\infty$ and $|\varGamma _{\textit{right}}|/DU_\infty$ , and corresponding ${\textit{Re}}_\varGamma$ , are shown in figure 13( a). Furthermore, the Reynolds number based on the vortex strength of the full symmetric vortex systems ${\textit{Re}}_{\varGamma _{\textit{total}}}$ is defined as

(5.1) \begin{equation} {\textit{Re}}_{\varGamma _{\textit{total}}}=\frac {\varGamma _{\textit{total}}}{\nu }, \end{equation}

where $\varGamma _{\textit{total}} = |\varGamma _{\textit{left}}| + |\varGamma _{\textit{right}}|$ . Figure 13( b) shows $\varGamma _{\textit{total}}/D U_\infty$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ where trends are displayed.

Figure 13. (a) $|\varGamma _{\textit{left}}|/DU_\infty$ , $|\varGamma _{\textit{right}}|/DU_\infty$ , and ${\textit{Re}}_{\varGamma }$ versus $x/D$ corresponding to flows over the non-spinning $\theta _c=10^\circ$ cone. (b) $\varGamma _{\textit{total}}/DU_\infty$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ computed from the corresponding $\varGamma _{\textit{left}}$ and $\varGamma _{\textit{right}}$ values plotted in panel (a).

The trends in $\varGamma _{\textit{left}}$ and $\varGamma _{\textit{right}}$ reveal a slight asymmetry, where $\varGamma _{\textit{left}}$ is consistently stronger than $\varGamma _{\textit{right}}$ . The asymmetry may be a result of a slight misalignment of the laser sheet and the $yz$ -plane. Inevitable variations in intensity within the laser sheets from the laser optics and discrepancies between the laser sheets generated from the two laser heads may also contribute to this asymmetry. Furthermore, the asymmetries could be attributed to slight asymmetry of the global flow field due to influences of the test section hardware. Nonetheless, the trends show a monotonic increase in $\varGamma _{\textit{left}}$ , $\varGamma _{\textit{right}}$ and $\varGamma _{\textit{total}}$ as $\alpha$ increases. Furthermore, monotonic increase in $\varGamma _{\textit{left}}$ , $\varGamma _{\textit{right}}$ and $\varGamma _{\textit{total}}$ is observed as $x/D$ increases. Hence, the consolidated values $\varGamma _{\textit{total}}$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ show that the vortex strengths increase as $\alpha$ and $x/D$ increase.

5.2.6. Asymmetric vortex systems

Spin breaks down the symmetry. The schematic of the asymmetric vortex system in figure 8(b) describes the general form of the vorticity contours of the $|S_D|\gt 0$ cases corresponding to $\alpha \gtrsim \theta _c$ . The vortex triad, whose primary vortex rotates in the same direction as $\varOmega$ , is defined as the cyclonic vortex triad and the opposite triad is defined as the anti-cyclonic vortex triad. In other words, the cyclonic vortex triad forms in the co-rotating meridian, while the anti-cyclonic vortex triad forms in the counter-rotating meridian. The general shapes of the triads are distorted from their symmetric arrangement in the non-spinning case and are generally shifted towards the direction of rotation. Note, in cases where $S_D$ is positive, the cyclonic primary and tertiary vortices are positively signed, whereas the anticyclonic primary and tertiary vortices are negatively signed.

The cyclonic vortex triad stretches around the co-rotating meridian; at distances sufficiently far enough from the vertex of the cone, its primary vortex crosses over into the counter-rotating meridian in close proximity of the anti-cyclonic primary vortex. The cyclonic primary vortex diminishes in cross-sectional size, while staying attached to the leeward surface; this may be a result of counter-flow between the primary vortex and the surface of the cone that is moving in the opposite direction. The cyclonic primary vortex is pulled beneath the anti-cyclonic primary vortex and spread over the surface of the cone as it continues to follow the direction of rotation. The effects of rotation alone may be sufficient in overcoming the flows induced by the cyclonic primary vortex. The secondary cyclonic vortex remains close to the primary vortex with reduced cross-sectional size; as the secondary vortex is generated by the induced down-wash of the primary vortex, its strength scales with the strength of its parent primary vortex; hence, it quickly vanishes with the effects of rotation. The tertiary vortex remains close to the surface of the cone; its major axis is stretched and its minor axis is shrunk, but it maintains its coherency along the full length of the cone before being leaving the field of view.

The anti-cyclonic vortex triad behaves differently; its primary vortex remains in the counter-rotating meridian over the full length of the cone and separates from the cone surface at some distance $x/D$ . Two criteria are used to identify the separation point of anticyclonic primary vortices. Separation is first identified when opposite-signed vorticity exists between the surface of the cone and the adjacent primary vortex. Second, once the primary vortex separates, the normal distance between the primary vortex and the surface of the cone monotonically increases with increasing $x/D$ . Using these criteria, the anti-cyclonic primary vortex detaches at $x/D \approx 1.70$ at $\alpha =30^\circ$ and $\alpha =36^\circ$ as estimated from the $\omega _x$ -contours corresponding to $S_D = 2$ in figures 11 and 12. Note that for $\alpha \leqslant 24^\circ$ , the anti-cyclonic vortex does not detach. These observations match those in the corresponding streakline images of figure 5.

Figure 12 allows for a direct side-by-side comparison between flows for $S_D = 2\text{ and } 3$ at the highest angle of incidence case, $\alpha = 36^\circ$ . The anti-cyclonic primary vortex detaches from the surface at $x/D\approx 1.70$ for $S_D=2$ (column 2) and $x/D\approx 1.36$ for $S_D=3$ (column 3); thus, an increase in $S_D$ expedites the detachment of the anti-cyclonic primary vortex. Furthermore, an increase in $S_D$ acts by pushing the detached portions of the anti-cyclonic primary vortex further away from the surface. In rows 4–6 corresponding to $x/D = 1.02, 1.36\text{ and }1.70$ of the same figures, sequential steps of the cyclonic primary vortex being pulled underneath the anti-cyclonic primary vortex appear to trigger the detachment of the anti-cyclonic primary vortex from the surface of the cone.

For $\alpha = 18^\circ \text{ to } 36^\circ$ , the cross-sectional size of the anti-cyclonic primary vortex grows along the $+x_*$ -direction. The flow between the primary vortex and the surface of the cone is unidirectional; the tangential directions of the leeward surface and the lower half of the primary vortex are aligned. This facilitated flow acts by augmenting the primary vortex, increasing its strength along the $+x_*$ -direction.

The strength of the anti-cyclonic primary vortices are defined as $\varGamma _{{anti\hbox{-}cyc}}$ and are computed using (2.5) over a rectangular area isolating the primary vortex. These are used to compute and plot non-dimensional values $|\varGamma _{{anti\hbox{-}cyc}}|/DU_\infty$ and the corresponding Reynolds number, ${\textit{Re}}_{\varGamma _{{anti\hbox{-}cyc}}}$ , in figure 14. Here, $\varGamma _{{anti\hbox{-}cyc}}$ begins to diminish at $x/D \gt 1.70$ , where the anti-cyclonic primary vortex begins to leave the field of view. Additionally, $\varGamma _{{anti\hbox{-}cyc}}$ corresponding to $S_D=2 \text{ and } 3$ both increase as $x/D$ increases; however, $\varGamma _{{anti\hbox{-}cyc}}$ corresponding to $S_D=3$ is consistently larger than the $S_D=2$ counterpart. This demonstrates that the anti-cyclonic vortex strength is directly influenced by the magnitude of $S_D$ ; moreover, an increase in $S_D$ results in a stronger anti-cyclonic vortex.

Notably, $\varGamma _{{anti\hbox{-}cyc}}$ corresponding to $S=3$ continues to grow in magnitude past the estimated detachment distance $x/D \approx 1.36$ . This increase is thought to result from a vortex merger event, which is observed and discussed at the end of § 5.2.3. Fortuitously, the snapshot corresponding to row 6 column 2 of figure 7 provides smoke visualisation for the same flow condition reviewed here.

Figure 14. $|\varGamma _{{anti\hbox{-}cyc}}|/DU_\infty$ and ${\textit{Re}}_{\varGamma _{{anti\hbox{-}cyc}}}$ versus $x/D$ corresponding to flows over the spinning $\theta _c=10^\circ$ cone at $\alpha =36^\circ$ and $S_D= 2$ and $3$ .

5.2.7. Summary of flows over the $\theta _c = 10^\circ$ cone

At zero incidence, qualitative agreement among potential flow streamlines, smoke streaklines and streamwise PIV streamlines builds confidence in experimental set-ups. At angles of incidence $\alpha \lesssim \theta _c$ , effects of rotation are generally confined to the boundary layer. For spinning cases, streaklines ingested by the boundary layer follow the direction of rotation and reveal counter-flows in the counter-rotating meridian. At angles of incidence $\alpha \gtrsim \theta _c$ , flow separation at the leeward surface is present for both the spinning and non-spinning cases, and detected in both smoke visualisation and velocity measurements. Vortical flows remain attached to the leeward surface of the cone up to a distance $x_*/D$ that monotonically decreases as $\alpha$ increases. Similarly, an increase in $S_D$ expedites flow detachment from the leeward surface.

For non-spinning cases at $\alpha \gtrsim \theta _c$ , a steady symmetric pair of vortex triads emerge from the cone vertex, grow in cross-sectional size along the $+x_*$ -direction and remain attached to the leeward surface of the cone up to the highest incidence case of $\alpha = 36^\circ$ . As $\alpha$ increases, the total vortex system strength, $\varGamma _{\textit{total}}$ , increases. The vortex triads consist of primary and secondary vortices of nearly circular cross-sections and a tertiary vortex with an elliptical cross-section. The primary and tertiary vortices share the same sign of vorticity, and the secondary forms in between them with opposite sign. The primary vortices form from detached shear layers in the separation regions near the leeward surface of the cone. As the primary vortices form, they are pushed towards the surface from their induced down-wash, where secondary vortices form from viscous effects.

Asymmetries of the flows over spinning cones are highlighted over the full range $\alpha$ , where cyclonic and anti-cyclonic vortex triads are characterised from $\omega _x$ -contours. The cyclonic vortex triad stretches around the co-rotating meridian; its primary vortex diminishes in cross-sectional size along the length of the cone, stays attached to the leeward surface, and gets pulled in the direction of rotation until it is sucked beneath the anti-cyclonic primary vortex and smeared along the surface. It is conjectured that the vortex strength disparity between the cyclonic and anti-cyclonic primary vortices allows the anti-cyclonic primary vortex to pull the cyclonic primary vortex underneath it. Once this happens, the anti-cyclonic primary vortex detaches from the surface of the cone. An increase in $S_D$ is observed to expedite detachment of the anti-cyclonic primary vortex. Furthermore, an increase in $S_D$ results in an increase in anti-cyclonic primary vortex strength.

6.1.1. Potential flow streamlines versus streaklines and PIV streamlines

The flows corresponding to $\alpha = 0$ and $S_D = 0$ are compared directly with potential flow theory to gauge the success of the experiments. The red $\psi$ streamlines are overlaid on the corresponding streakline image in figure 16(a) and black $\psi$ streamlines are overlaid on the $|\boldsymbol{u}|/U_\infty$ contour plot and white streamlines in figure 16(b). Good agreement among the potential flow streamlines, smoke streaklines and PIV streamlines is observed. Better agreement with the PIV streamlines is a result of more precise location of the plane of symmetry with laser sheets.

Figure 16. $\psi (r,\theta )$ streamlines overlaid on streaklines and PIV streamlines at $y\approx 0$ corresponding to $\theta _c=15^\circ$ , $S_D=0$ and $\alpha =0$ .

6.1.2. Streakline behaviours near the surface

The streakline images corresponding to $\alpha \lesssim \theta _c$ (rows 1–3 of figure 15) exhibit similar patterns depicted by the similar set of streakline images corresponding to $\theta _c = 10^\circ$ in figure 5 (rows 1–2). Flows corresponding to $\alpha \leqslant 6^\circ$ remain close to the surface of the cone, where streaklines outside of the boundary layer region are unaffected by rotation and are nearly identical between non-spinning and spinning cases. Streaklines ingested by the boundary layer in $|S_D|\gt 0$ cases exhibit effects of $\varOmega$ as they are curved in the direction of rotation.

Details of streaklines near the surface corresponding to $\alpha = 12^\circ$ (row 3) show interesting behaviour of the streaklines. For the non-spinning case, streaklines emerging from the windward meridian are shaped by the surface of the cone; as they approach the leeward surface, they align parallel with it. Streaklines in the $S_D=+2$ case are pulled significantly in the direction of rotation. Streaklines in the $S_D=-2$ case highlight counter-flows in the counter-rotating meridian. A region devoid of streaklines directly above the leeward surface emerges at $\alpha =12^\circ$ , marking the beginnings of flow separation; hence, $\alpha _*\approx 12^\circ$ for $\theta _c=15^\circ$ , and $\varLambda \approx 0.8$ . The effects of rotation appears to shrink the region devoid of streaklines in the $|S_D|\gt 0$ cases.

6.1.3. Vortical flows

Helical streakline patterns shaped by the trailing vortex systems near the leeward surface emerge at $\alpha = 18^\circ$ (row 4 of figure 15). The interior details of vortex systems revealed in non-spinning cases over the $\theta _c=15^\circ$ cone are more visible. Figure 17 labels the interior vortices of the camera-side symmetric vortex triad captured for $\theta _c=15^\circ$ , $\alpha = 24^\circ$ and $S_D=0$ (row 5, column 1 of figure 15). In this instance, signatures of primary, secondary and tertiary vortices are visible and the labels allow for a direct correspondence to the PIV analyses in § 6.2. Recall that the terms primary, secondary and tertiary vortices are defined in figure 8(a). Similar to the flows in § 5 for $\theta _c=10^\circ$ , the symmetric vortex system for $S_D=0$ remains close to the leeward surface of the cone over the full range of $\alpha$ , where the cross-sectional size of the vortex system grows as $\alpha$ increases.

Figure 17. Labelled symmetric vortex triad corresponding to $\theta _c=15^\circ$ , $S_D=0$ and $\alpha = 24^\circ$ .

The addition of spin introduces new complexities to the vortical flows near the leeward surface. Cyclonic vortices in the co-rotating meridian are visible in the $S_D=2$ cases, particularly for $\alpha = 18^\circ , 24^\circ , 30^\circ \text{ and }36^\circ$ (rows 4–7, column 2 of figure 15). In contrast to their behaviours in flows over the $\theta _c = 10^\circ$ cone, the cyclonic vortices detach from the surface at some distance $x/D$ and remain in the field of view. As $\alpha$ increases from $18^\circ$ to $36^\circ$ , the cyclonic vortices detach at decreasing distances $x/D$ . Detached portions of the cyclonic primary vortices monotonically lose their coherency in the $+x$ -direction and are marked by small-scale wave patterns.

The small-scale wave patterns appear to originate from wavy rolls at the leeward surface. Figure 18 labels the cyclonic primary vortex corresponding to $\alpha = 24^\circ$ and $S_D = 2$ , where small-scale rolls forming at the leeward surface are marked by magenta coloured dots. Streaklines originating from the roll features extend through the region underneath the detached cyclonic vortex and wrap around it. These features merge with the cyclonic primary vortex and likely mark cross-sections of spiral wave vortices forming within the boundary layer. The details of this mechanism are not fully tractable due to resolution constraints of the flow visualisation experiments; thus, these observations are made by conjecture. Nonetheless, the speculations merit further investigation of the boundary layer region and its interaction with the trailing vortices.

Figure 18. Labelled cyclonic primary vortex corresponding to $\theta _c=15^\circ$ , $S_D=2$ and $\alpha = 24^\circ$ . Magenta coloured dots mark small-scale rolls forming at the leeward surface.

Anti-cyclonic primary vortices are visualised in the $S_D=-2$ cases corresponding to $\alpha = 18^\circ , 24^\circ , 30^\circ \text{ and }36^\circ$ (rows 4–7, column 3 of figure 15); figure 19 shows a representative case corresponding to $\alpha =36^\circ$ and $S_D=-2$ (row 7, column 3). Within the same image, streaklines shaped by the cyclonic primary vortex in the co-rotating meridian are also labelled. Generally, the anti-cyclonic primary vortex follows the direction of rotation and remains close to the counter-rotating meridian of the cone. The anti-cyclonic primary vortices burst at sufficient distance from the cone vertex; this distance appears to increase as $\alpha$ increases. Bursting is identified here subjectively as a sudden enlargement over twice the size of the circumferential signature of a vortex and as loss of coherence in the streakline bundle. This behaviour is distinct from the anti-cyclonic vortices of flows over the cone of $\theta _c = 10^\circ$ , where they readily separate from the surface and remain suspended in regions above the leeward surface.

Figure 19. Labelled anti-cyclonic primary vortex corresponding to $\theta _c=15^\circ$ , $S_D=-2$ and $\alpha = 36^\circ$ . The cyclonic primary vortex in the co-rotating meridian is also visible and labelled.

6.2.1. Oblique measurements

The $\omega _xD/U_\infty$ -contours corresponding to non-spinning cases at angles of incidence $\alpha \gtrsim \theta _c$ show the same topology depicted in figure 8(a); they are characterised by a symmetric pair of vortex triads consisting of primary, secondary and tertiary vortices. However, vorticity contours corresponding to flows over a spinning cone of half-angle $\theta _c=15^\circ$ exhibit different configurations from those sketched in figure 8(b). Figure 20 shows a schematic characterising the asymmetric vortex systems on a half-angle cone $\theta _c=15^\circ$ ; it summarises the topology of oblique $\omega _x$ -contours corresponding to $S_D=2$ cases, where the asymmetric vortex arrangements are distinguished mainly by the behaviours of the cyclonic and anti-cyclonic primary vortices.

Figure 20. Schematic of the asymmetric trailing vortex systems behind a spinning cone of half-angle $\theta _c = 15^\circ$ . Cyclonic vortex triads are defined in the co-rotating meridian, and anti-cyclonic vortex triads in the counter-rotating meridian.

Figure 21. $\omega _xD/U_\infty$ in $yz$ -planes corresponding to $\theta _c=15^\circ$ , $S_D=0$ and $\pm 2$ , and $\alpha = 36^\circ$ .

Figure 21 shows $\omega _xD/U_\infty$ -contours with overlaid streamlines coloured in magenta in $yz$ -planes at $x/D = 0 {-} 1.56$ in 0.225 increments corresponding to $S_D = 0 \text{ and} \pm 2$ and $\alpha = 36^\circ$ for comparison between $|S_D|\gt 0$ cases in the oblique measurement plane. The symmetric vortex systems highlighted by both the streakline images of figure 15 and the oblique PIV measurements of figure 21 grow in cross-sectional size and vortex strength along the length of the cone.

Figure 22( a) shows a plot of the non-dimensional form $|\varGamma _{{left},right}|/DU_\infty$ and the corresponding ${\textit{Re}}_{\varGamma }$ , and figure 22( b) shows a plot of $\varGamma _{\textit{total}}/DU_\infty$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ versus $x/D$ . Each curve corresponds to some $\alpha$ . The differences between $|\varGamma _{\textit{right}}|$ and $|\varGamma _{\textit{left}}|$ adequately demonstrate the expected symmetry. The strengths of the vortex triads are shown to increase along the length of the cone as they are continuously fed by the surface-attached shear layer. In addition, they are shown to increase monotonically as $\alpha$ increases.

Figure 22. (a) $|\varGamma _{\textit{left}}|/DU_\infty$ , $|\varGamma _{\textit{right}}|/DU_\infty$ and ${\textit{Re}}_{\varGamma }$ versus $x/D$ corresponding to flows over the non-spinning $\theta _c=15^\circ$ cone. (b) $\varGamma _{\textit{total}}/DU_\infty$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ computed from the corresponding $\varGamma _{\textit{left}}$ and $\varGamma _{\textit{right}}$ values plotted in panel (a).

For spinning cases, the cyclonic primary vortex detaches from the leeward surface of the cone and progressively shifts further from the surface of the cone along the $+x$ -direction. The first instance this behaviour is detectable in the oblique measurements corresponds to $\alpha =24^\circ$ ; as $\alpha$ increases, the cross-sections become increasingly visible. The cyclonic primary vortex is pushed in the direction of rotation, but never crosses over into the the counter-rotating meridian; this behaviour is distinct from that observed over the more slender $\theta _c=10^\circ$ cone.

The vortex strengths associated with the cyclonic and anti-cyclonic primary vortices, $\varGamma _{{cyc}}$ and $\varGamma _{{anti\hbox{-}cyc}}$ , are computed from $\omega _x$ -contours corresponding to $\alpha =36^\circ$ and $S_D=\pm 2$ (columns 2 and 3 of figure 21). The non-dimensional magnitudes are plotted in figure 23. Again, the integration regions corresponding to $\varGamma _{{cyc}}$ and $\varGamma _{{anti\hbox{-}cyc}}$ only encapsulate regions containing the primary vortices of interest. The values corresponding to $S_D=\pm 2$ demonstrate the expected symmetry, i.e. $|\varGamma _{{anti\hbox{-}cyc}}|/D U_\infty$ and $|\varGamma _{{cyc}}|/D U_\infty$ corresponding to $S_D=\pm 2$ are nearly coincident. Symmetry is also apparent from qualitative inspection of the corresponding $\omega _x$ -contours (figure 21).

Figure 23. $|\varGamma _{{cyc}}|/DU_\infty$ and $|\varGamma _{{anti\hbox{-}cyc}}|/DU_\infty$ versus $x/D$ corresponding to flows over the spinning $\theta _c=15^\circ$ cone at $\alpha =36^\circ$ and $S_D=\pm 2$ .

At sufficient distance from the vertex of the cone, detached portions of the cyclonic primary vortex either decay in strength or remain the same as depicted by qualitative inspection of the vorticity contours along the $+x$ -direction. This observation is supported by $\varGamma _{{cyc}}$ computations as $|\varGamma _{{cyc}}|$ for both $S_D=\pm 2$ cases monotonically increase along the $+x$ -direction until $x/D \approx 1.12$ , where the vortex strengths begins to stagnate. The reason for the vortex strength stagnation period is as follows: once the cyclonic primary vortex fully detaches from the leeward surface of the cone, it detaches from the surface-attached shear layer and is essentially cutoff from its vorticity supply. Furthermore, counter-flow regions between the cyclonic primary vortex and the surface of the cone work against the direction of rotation of the cyclonic primary vortex; the tangential flow of the lower half of the vortex and the surface of the cone are in opposing directions.

The behaviour of the anti-cyclonic primary vortex for $\theta _c = 15^\circ$ embraces the surface of the cone and grows in cross-sectional size and vortex strength along the $+x$ -direction. This is a result of the anti-cyclonic primary vortex being continuously fed by vorticity from the adjacent shear layer. Furthermore, the tangential velocity of the leeward surface of the cone and the bottom portion of the anti-cyclonic primary vortex are unidirectional; hence, the flow within this region is facilitated. The corresponding $\varGamma _{{anti\hbox{-}cyc}}$ values corresponding to the $S_D=\pm 2$ cases support this conclusion; in both cases, $|\varGamma _{{anti\hbox{-}cyc}}|$ increases along the $+x$ -direction. At $x/D \gtrsim 1.12$ , $\varGamma _{{anti\hbox{-}cyc}}$ starts to decay as the vortex begins to leave the field of view. The anti-cyclonic vortices are suspected to burst at sufficient distance from the vertex of the cone, as observed in the $S_D=-2$ streakline images of figure 15; however, this behaviour is not captured within the field of view of the oblique PIV measurements.

6.2.2. Isosurfaces of trailing vortex systems

Isometric views of vortices are constructed from a series of oblique plane vorticity measurements made at $x/D = 0 - 1.78$ in 0.05 increments for $\theta _c = 15^\circ$ , $S_D = 0$ , 2 and 3, and $\alpha = 30^\circ$ . Figure 24( a) corresponds to $S_D = 0$ , and figures 24( b) and 24(c) correspond to $S_D=2$ and $S_D=3$ , respectively. The $\omega _x$ -contours are used to generate isosurfaces, where values between the measurement planes are interpolated with a cubic fit. Red isosurfaces denote $\omega _x\gt 0$ and blue denote $\omega _x\lt 0$ . The transparency of the isosurfaces increase with decreasing vorticity magnitudes. Therefore, large vorticity magnitudes correspond to opaque isosurfaces near the vortex centres and weaker regions correspond to more transparent isosurfaces further away. No vorticity scale is provided for the isosurface plots, as they mainly serve as a qualitative resource to provide a visualisation of the trailing vortex systems from behind the cone. The viewing angle of the isosurfaces corresponds to a line of view behind the cone that is aligned with the central axis of the wind tunnel. Sketches of the symmetric vortex system depicted in figure 8(a) are overlaid on the cross-sections of the vortex triads near the base of the cone ( $x/D = 1.78$ ) for illustrative purposes. Similarly, the asymmetric vortex system depicted in figure 20 is overlaid on the cross-sections of the vortex system at $x/D = 1.78$ in figures 24( b) and 24(c).

Figure 24. $\omega _x$ -isosurface plots corresponding to $\theta _c = 15^\circ$ , $\alpha = 30^\circ$ , and $S_D =0,\, 2\text{ and }\,3$ . Sketches of the vortex systems are overlaid on the vortex cross-sections near the base of the cone for illustrative purposes. (a) $S_D=0$ , (b) $S_D=2$ , (c) $S_D=3$ .

The anti-cyclonic vortex triad is shifted outside the field of view as it is pulled in the direction of rotation for both figures 24( b) and 24(c); hence, the location and scale of the associated cross-sectional sketches are rough estimates. The cyclonic primary vortex for $S_D=3$ lies further away from the leeward surface of the cone at $x/D=1.78$ when compared with the $S_D=2$ plot. The cyclonic primary vortex in the $S_D=2$ plot appears less transparent, which is associated with stronger vorticity magnitude when compared with the $S_D=3$ counterpart. Therefore, an increase in angular rotation, $\varOmega$ or $S_D$ , expedites the separation of the cyclonic primary vortex from the leeward surface of the cone; thus, the cyclonic primary vortex corresponding to $S_D=3$ is cut off from vorticity of the surface-attached shear layer at a shorter distance $x/D$ , allowing it longer distances to decay.

6.2.3. Summary of flows over the cone of half-angle $\theta _c = 15^\circ$

Flows over the $\theta _c = 15^\circ$ cone resemble those over the $\theta _c = 10^\circ$ cone at angles of incidence $\alpha \lesssim \theta _c$ . Vortical flows corresponding to $S_D=0$ and $\alpha \gtrsim \theta _c$ exhibit the same behaviours of symmetric vortex triads over the $\theta _c = 10^\circ$ cone. In contrast, distinct behaviours of the asymmetric vortex systems are observed for $|S_D|\gt 0$ cases at angles of incidence $\alpha \gtrsim \theta _c$ .

The cyclonic primary vortex is pushed in the direction of rotation, but does not cross the symmetry plane. At sufficient distances from the vertex of the cone, it detaches from the leeward surface and ceases to grow in strength.

The mechanism for the altered behaviour of the primary cyclonic vortex is as follows: the primary cyclonic vortex forms in the co-rotating meridian and is pushed in the direction of rotation through counter-flow effects between the vortex and the leeward surface of the cone; at sufficient distance $x/D$ from the vertex, depending on $\alpha$ and $S_D$ , the vortex detaches from the surface before making it past the plane of symmetry. Once detached, the vortex is no longer fed by the surface-attached shear layer; hence, it ceases to grow in strength and eventually begins to decay. The detachment distance $x/D$ is observed to decrease with increasing $S_D$ or $\alpha$ .

The anti-cyclonic primary vortex also shifts in the direction of rotation along the $x/D$ direction; however, it remains attached to the surface of the cone in the counter-rotating meridian until it bursts; it abruptly collapses and is destroyed along the surface. As the anti-cyclonic primary vortex wraps around the counter-rotating meridian of the cone, it grows in size and vortex strength as it is continuously fed by the surface-attached shear layer and facilitated by the co-flow region between the vortex and the surface of the cone. The anti-cyclonic primary vortex grows in strength as $S_D$ increases.

Small-scale rolls mark streaklines near the leeward surface of $|S_D|\gt 0$ cases at angles of incidence $\alpha \gtrsim \theta _c$ ; they likely reveal signatures of the expected centrifugal boundary layer instability taking the form of spiral waves. However, due to temporal and spatial resolution constraints of the experiments, no comprehensive analyses were conducted on the boundary layer regions.

7.1.1. Potential flow streamlines versus streaklines and PIV streamlines

The $\psi$ -contours for the $\theta _c=22.5^\circ$ cone are overlaid the streaklines corresponding to $S_D=0$ and $\alpha =0$ in figure 27(a). The agreement with potential flow streamlines are sufficient, with better agreement observed near the vertex of the cone. Discrepancy between potential flow streamlines and streaklines near the right side of the field of view is a result of smoke streaklines leaving the plane of symmetry, a reoccurring difficulty in visualising the plane of symmetry with smoke streaklines. Close agreement is observed between potential flow streamlines and the streamwise PIV streamlines (in white) overlaid in black on the $|\boldsymbol{u_*}|/U_\infty$ contour plot in figure 27(b). Again, the agreement between the PIV counterpart is consistently better as a result of more precise measurement of the plane of symmetry with fixed laser sheets of the PIV experiment.

Figure 27. $\psi (r,\theta )$ streamlines overlaid streaklines and PIV streamlines at $y\approx 0$ corresponding to $\theta _c=22.5^\circ$ , $S_D=0$ and $\alpha =0$ .

7.1.2. Streaklines behaviours near the surface

Streaklines ingested by the boundary layer are shaped in the direction of rotation corresponding to angles of incidence $\alpha \lesssim 12^\circ$ , rows 1–3 of figures 25 and 26, while streaklines outside of the boundary layer regions are unaffected by rotation. As expected, the streaklines ingested by the boundary layer for $|S_D|=2$ cases of figure 26 are more tightly curved in the direction of rotation in comparison to the $S_D=\pm 1$ cases of figure 25. Streaklines that emerge from the co-rotating meridian and curve into the counter-rotating meridian are visible in columns associated with $S_D\lt 0$ (column 3 of figures 25 and 26); these streaklines allow for visualisation of counter-flows in the counter-rotating meridian. As $\alpha$ is increased, asymmetries of streaklines near the surface monotonically increase.

Unique details are uncovered by the streaklines corresponding to $S_D=-1$ and $\alpha = 18^\circ$ (row 4, column 3 of figure 25); a close-up view corresponding to a cropped area of the streakline image is shown in figure 28( a). Cross-hatch patterns form between the streaklines emerging from the co-rotating meridian deep inside of the boundary layer and the streaklines following the surface curvature at the counter-rotating meridian; thus, highlighting regions of counter-flow. Small-scale rolls are observed near the leeward surface, which are also present in the $S_D\gt 0$ streakline images associated with $\alpha \geqslant 18^\circ$ . Regularly spaced angled wave patterns mark the streaklines near the windward surface of the cone, which appear to have the same wavelength/spacing of the wavy rolls at the leeward surface. The wave patterns at the windward surface and small-scale rolls are likely signatures of inter-related cross-flow and centrifugal boundary layer instabilities.

A close-up view of the streakline image corresponding to $\theta _c=22.5^\circ$ , $S_D=-1$ and $\alpha = 24^\circ$ is shown in figure 28( b). Streaklines ingested by the rotating boundary layer following the direction of rotation are distinctly marked by sinusoidal wave patterns, signatures of spiral-wave centrifugal instabilities, the spacing of which resembles those of Tambe et al. (Reference Tambe, Schrijer and Veldhuis2021).

At the highest case corresponding to $\alpha = 36^\circ$ , $S_D=1$ and $S_D=2$ , angled wave patterns are detected on the streaklines near the windward surface of the cone and extend around the entire co-rotating surface. Figure 28( c) provides a close-up view of the wave patterns marking the streaklines near the surface corresponding to $S_D=1$ and $\alpha =36^\circ$ of figure 25. It is unclear from the streakline patterns alone; however, the wave patterns may be a result of cross-flow instability within the boundary layer.

Figure 28. Close-up views of streakline images corresponding to $\theta _c=22.5^\circ$ . (a) Angled wave patterns emerge on streaklines near the windward surface, and bead-like features emerge near the leeward surface. $S_D=-1$ and $\alpha = 18^\circ$ . (b) Streaklines highlighting regions of counter-flow are marked by wave patterns. $S_D=-1$ and $\alpha = 24^\circ$ (c) Angled wave patterns mark the streaklines along the entire surface of the cone. $S_D=1$ and $\alpha = 36^\circ$ .

7.1.3. Vortical flows

Flows corresponding to $\theta _c=22.5^\circ$ and $\alpha \geqslant 18^\circ$ exhibit similar helical features marked by the trailing vortex systems of previous sections, and hence, $\alpha _* \approx 18^\circ$ and $\varLambda \approx 0.8$ for $\theta _c=22.5^\circ$ . Symmetric vortex triads consisting of primary, secondary and tertiary vortices form on the leeward surface corresponding to $S_D=0$ and $\alpha \gtrsim \theta _c$ . Figure 29( a) identifies the helical structures forming over the $S_D=0$ and $\alpha =30^\circ$ case (row 6, column 1 of figure 25), where interiors of the vortex triad are visible. The cross-sectional size of the symmetric vortex systems grow as $\alpha$ increases.

Figure 29. Labelled vortex systems corresponding to $\theta _c=22.5^\circ$ : (a) $S_D=0$ and $\alpha = 30^\circ$ ; (b) $S_D=2$ and $\alpha = 24^\circ$ ; (c) $S_D=-2$ and $\alpha = 36^\circ$ .

The primary cyclonic vortex is visible near the leeward surface for $S_D\gt 0$ cases (column 2 of figures 25 and 26). Mimicking the behaviours of the cyclonic primary vortices of § 6, they emerge from the cone vertex and detach from the leeward surface; as $\alpha$ increases, the separation distance along the $x_*$ -direction decreases. Detached portions of cyclonic vortices are marked by the same small-scale rolls observed in § 6; signatures of the centrifugal instabilities of the rotating boundary layer. Figure 29( b) labels the cyclonic primary vortex visible in the streakline image corresponding to $\theta _c=22.5^\circ$ , $S_D=2$ and $\alpha = 24^\circ$ , where regularly spaced, small-scale rolls are marked by magenta dots. The rolls are visible with variable clarity in each of the $S_D\gt 0$ flows for $\alpha \geqslant 18^\circ$ .

Helical patterns shaped by the anti-cyclonic primary vortices in the counter-rotating meridian are visible in $S_D\lt 0$ cases. The behaviours mimic those described in § 6, where the anti-cyclonic primary vortices embrace the surface as they are pushed in the direction of rotation; which burst at some downstream from the vertex. Figure 29( c) labels the anti-cyclonic primary vortex visible in the streakline image corresponding to $\theta _c=22.5^\circ$ , $S_D=-2$ and $\alpha =36^\circ$ (row 7, column 3 of figure 26); the cyclonic primary vortex in the co-rotating meridian is also visible and labelled within the figure.

Figure 30. $\omega _xD/U_\infty$ in $yz$ -planes corresponding to $\theta _c=22.5^\circ$ , $S_D\in [0,2]$ and $\alpha = 36^\circ$ .

7.2.1. Oblique measurements

Figure 30 corresponding to $\theta _c=22.5^\circ$ , $S_D = 0 \text{ and } 2$ , and $\alpha = 36^\circ$ shows $\omega _xD/U_\infty$ -contours with overlaid magenta streamlines computed from measurements at $x/D = 0 - 1.01$ in 0.145 increments. In each figure, the rows correspond to $x/D$ and the columns to $S_D$ . The $\omega _xD/U_\infty$ -contours for $\theta _c = 22.5^\circ$ and $S_D=0$ have similar topology as the more slender $\theta _c = 15^\circ$ cone; they are characterised by a symmetric pair of vortex triads that grow along the $+x$ -direction. However, the symmetric vortex systems are more stretched along the leeward surface depicted by the increased lateral spacing between the triads.

Here, $|\varGamma _{\textit{left}}|/DU_\infty$ , $|\varGamma _{\textit{right}}|/DU_\infty$ and ${\textit{Re}}_{\varGamma }$ versus $x/D$ are plotted in figure 31( a) for $\alpha \in [21^\circ ,24^\circ ,30^\circ ,36^\circ ]$ . Additionally, $\varGamma _{\textit{total}}/DU_\infty$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ are plotted in the same manner in figure 31( b). As before, the symmetric vortex systems grow in strength in the $+x$ -direction and as $\alpha$ increases, the vortex strengths increase. Generally, symmetry is preserved between $\varGamma _{\textit{right}}$ and $\varGamma _{\textit{left}}$ .

The oblique $\omega _x$ -contours corresponding to $S_D=2$ demonstrate the same behaviours characterised in figure 20. The anti-cyclonic primary vortex shifts in the direction of rotation along the length of the cone and remains attached to surface until it leaves the field of view; for $\theta _c=22.5^\circ$ , the anti-cyclonic pair completely leaves the field of view at $x/D \approx 0.87$ . The cyclonic primary vortex detaches from the leeward surface and begins to stagnate or decay in strength with distance from the vertex. In contrast to the more slender cones of previous sections, the cyclonic primary vortex is pulled in the direction of rotation to a lesser extent. The counter-flow regions between the cyclonic primary vortex and the surface of the cone seem to have more significant effects that weaken $\varGamma _{{cyc}}$ as $\theta _c$ increases. Furthermore, distances where the cyclonic primary vortex detaches from the leeward surface are estimated from the contour plots. This is most clearly depicted in the contours associated with $\alpha =36^\circ$ and $S_D=2$ , where the cyclonic primary vortex detaches at $x/D\approx 0.58$ .

Figure 31. (a) $|\varGamma _{\textit{left}}|/DU_\infty$ , $|\varGamma _{\textit{right}}|/DU_\infty$ and ${\textit{Re}}_{\varGamma }$ versus $x/D$ corresponding to flows over the non-spinning $\theta _c=22.5^\circ$ cone. (b) $\varGamma _{\textit{total}}/DU_\infty$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ computed from the corresponding $\varGamma _{\textit{left}}$ and $\varGamma _{\textit{right}}$ values plotted in panel (a).

The $\varGamma _{{cyc}}$ and $\varGamma _{{anti\hbox{-}cyc}}$ corresponding to $\theta _c=22.5^\circ$ , $S_D=2$ and $\alpha = 36^\circ$ are plotted in figure 32. The missing values at $x/D = 0$ and 0.14 are due to the inability to distinguish there the primary vortex from the tertiary. $\varGamma _{{anti\hbox{-}cyc}}$ at $x/D = 0.87$ and 1.01 are missing as the anti-cyclonic primary vortex has left the field of view. The same trends are seen in both primary vortices as described in § 6.2. The cyclonic primary vortices begin to decay/stagnate at sufficient distances from the vertex; here, the detachment distance is estimated as $x/D \approx 0.72$ . The anti-cyclonic primary vortex is shown to grow in strength along the $x/D$ -direction until it leaves the field of view.

Figure 32. $|\varGamma _{{cyc}}|/DU_\infty$ and $|\varGamma _{{anti\hbox{-}cyc}}|/DU_\infty$ versus $x/D$ corresponding to flows over the spinning $\theta _c=22.5^\circ$ cone at $\alpha =36^\circ$ and $S_D=2$ .

8.1.1. Potential flow streamlines versus streaklines and PIV streamlines

The $\psi$ -contours are overlaid streaklines corresponding to $S_D=0$ and $\alpha =0$ in figure 35(a), where good agreement with potential flow streamlines is observed over the full field of view of the streakline image. However, disparities are evident with streaklines near the leeward surface; a likely result of the streaklines leaving the plane of symmetry. Better agreement is shown in the overlaid potential flow streamlines corresponding to $|\boldsymbol{u_*}|/U_\infty$ contours and streamlines in figure 35(b). Better agreement with the PIV streamlines are attributed to more precise alignment of the laser sheets with the $y=0$ plane.

Figure 35. $\psi (r,\theta )$ streamlines overlaid streaklines and PIV streamlines at $y\approx 0$ corresponding to $\theta _c=30^\circ$ , $S_D=0$ and $\alpha =0$ .

8.1.2. Streakline behaviours near the cone surface

The streakline patterns over a $\theta _c=30^\circ$ cone are consistent with the patterns of $\theta _c = 15^\circ$ and $22.5^\circ$ cones, with a few subtleties. Flows outside of the boundary layer region corresponding to $\alpha \lesssim \theta _c$ are unaffected by rotation, where streaklines that come in close proximity are ingested by the rotating boundary layer and shaped in the direction of rotation. Streakline images corresponding to $\alpha \leqslant 18^\circ$ , rows 1–4 of figures 33 and 34, show with increasing $S_D$ noticeably more arched curves form in the ingested streaklines.

Similar to flows over the cone of $\theta _c = 22.5^\circ$ , the additional light source used for streakline images corresponding to the $S_D=0\,\text{ and }\pm 1$ cases in figure 33 allow for enhanced visualisation of the streaklines near the surface. Regularly spaced, angled wave patterns on streaklines near the surface and wavy rolls near the leeward surface that persist in streakline images of $\theta _c=22.5^\circ$ are also detected here. Figure 36 shows close-up portions focused on areas near the surface of the streakline images corresponding to $\alpha = 24^\circ$ and $S_D=0\,\text{and}\pm 1$ (row 5 of figure 33); wave patterns along the surface of the cone are present in each of the $S_D=0\text{ and}\pm 1$ cases. As wave patterns exist in non-spinning cases, they cannot be a result of centrifugal instability; thus, they indicate cross-flow instability. Nonetheless, rolls are present in both $S_D =\pm 1$ cases, further supporting the argument that they are signatures of centrifugal spiral-wave instabilities.

Figure 36. Close-up sections focused on streaklines near the surface corresponding to $\theta _c=30^\circ$ , $S_D=0\,\text{ and}\pm 1$ , and $\alpha = 24^\circ$ . (a) $S_D=0$ , (b) $S_D=1$ , (c) $S_D=-1$ .

The same wavy rolls are present near the leeward surface of all $|S_D|\gt 0$ cases corresponding to $\alpha \gtrsim \theta _c$ . As $\alpha$ increases past $\theta _c$ , the rolls imprint their signatures on the adjacent trailing vortices. Close-up sections focused on streaklines near the surface corresponding to $\theta _c=30^\circ$ , $S_D = \pm 1$ and $\alpha = 30^\circ$ are shown in figure 37. The $S_D=1$ case shows distinct small-scale rolls along the leeward surface, where their influence on the trailing cyclonic vortex is apparent. Wave patterns are detected along the entire surface of the cone that form with the same spacing/frequency as the wavy rolls, which implies a connection between them. Moreover, the $S_D=-1$ case highlights counter-flow regions in the counter-rotating meridian that are marked by cross-hatch patterns. Streaklines ingested by the boundary layer highlight the direction of rotation and they are distinctly marked by small-scale wave patterns. Rolls near the leeward surface corresponding to $S_D=-1$ form with consistent spacing detected in the $S_D=1$ counterpart; hence, demonstrating the capability to visualise the same phenomena from both viewing angles.

Figure 37. Close-up sections focused on streaklines near the surface corresponding to $\theta _c=30^\circ$ , $S_D = \pm 1$ and $\alpha = 30^\circ$ . (a) $S_D=1$ , (b) $S_D=-1$ .

8.1.3. Vortical flows

In both cases, streaklines near the leeward surface for $\alpha \leqslant 12^\circ$ are aligned nearly parallel to the surface; however, at angles of incidence $\alpha \gtrsim 18^\circ$ , a region devoid of streaklines forms directly above it. The void regions depict induced down-wash effects; at $\alpha \gt 18^\circ$ , helical patterns shaped by the trailing vortices emerge at the leeward surface. The separation angle is estimated as $\alpha _* \approx 21^\circ$ ; hence, for the cone $\theta _c = 30^\circ$ , $\varLambda \approx 0.7$ .

The helical patterns shaped by the trailing vortices over non-spinning cones are similar to those corresponding to $\theta _c = 15^\circ \text{ and } 22.5^\circ$ . Figure 38 shows a representative case of the vortical flows associated with $\theta _c=30^\circ$ , $S_D=0 \text{ and }\pm 2$ , and $\alpha =36^\circ$ (row 7 of figure 34); figure 38(a) labels the symmetric vortex system in the $S_D=0$ streakline image, figure 38(b) labels the cyclonic primary vortex in the $S_D=2$ streakline image, and figure 38(c) labels the anti-cyclonic primary vortex, cyclonic primary vortex and areas of cross-flow near the counter-rotating surface of the $S_D=-2$ streakline image.

Figure 38. Labelled vortex systems corresponding to $\theta _c = 30^\circ$ , $S_D=0\,\mathrm{and}\,\pm 2$ and $\alpha =36^\circ$ . (a) Labelled vortex triad composed of primary, secondary and tertiary vortices, $S_D=0$ . (b) Labelled cyclonic primary vortex, $S_D=2$ . (c) Labelled cyclonic primary vortex, $S_D=2$ . Labelled anti-cyclonic primary vortex, cyclonic primary vortex in the co-rotating meridian and regions of cross-flow near the counter-rotating meridian surface, $S_D=-2$ .

Vortices over non-spinning cases form nearly parallel to the surface of the cone and grow in size along the $+x$ -direction. In certain cases, the interiors of the symmetric vortex triad are visible and show signatures of primary, secondary and tertiary vortices. Furthermore, as observed for the $\theta _c=15^\circ$ and $22.5^\circ$ cases, the cyclonic primary vortex appears in $S_D\gt 0$ -cases, where it detaches from the leeward surface of the cone and remains in the field of view. The anti-cyclonic primary vortices are also depicted in $S_D\lt 0$ -cases, where they appear to wrap closely around the counter-rotating surface until they burst and form puffs of turbulent smoke in regions downstream. The anti-cyclonic primary vortices burst at shorter distances along the $+x$ -direction when compared with the corresponding $\theta _c=15^\circ$ and $22.5^\circ$ flows.

8.2.1. Oblique measurements

Figure 39 shows $\omega _xD/U_\infty$ -contours with overlaid streamlines corresponding to $\theta _c=30^\circ$ , $S_D = 0\text{ and }+2$ , and $\alpha = 36^\circ$ at $x/D = 0 - 0.72$ in $0.1$ increments.

Figure 39. $\omega _xD/U_\infty$ in $yz$ -planes corresponding to $\theta _c=30^\circ$ , $S_D\in [0,2]$ and $\alpha = 36^\circ$ .

Vortex strengths corresponding to figure 39 are summarised in figures 40( a) and 40(b). In agreement with previous sections, the symmetric vortex systems grow in strength along the length of the cone and as $\alpha$ increases. Symmetry in the trailing vortex systems corresponding to $S_D=0$ is qualitatively depicted by the corresponding $\omega _x$ -contours. Moreover, symmetry is conserved between $\varGamma _{\textit{right}}$ and $\varGamma _{\textit{left}}$ , as observed in figure 40( a). The symmetric vortex systems are stretched further along the leeward surface; hence, they are more shrunk in the $z_*$ direction. Spacing between the vortex triads are larger than those depicted over a cone of half-angle $\theta _c=22.5^\circ$ .

Figure 40. (a) $|\varGamma _{\textit{left}}|/DU_\infty$ , $|\varGamma _{\textit{right}}|/DU_\infty$ and ${\textit{Re}}_{\varGamma }$ versus $x/D$ corresponding to flows over the non-spinning $\theta _c=30^\circ$ cone. (b) $\varGamma _{\textit{total}}/DU_\infty$ and ${\textit{Re}}_{\varGamma _{\textit{total}}}$ computed from the corresponding $\varGamma _{\textit{left}}$ and $\varGamma _{\textit{right}}$ values plotted in panel (a).

The $\omega _x$ -contours corresponding to $S_D=2$ agree with the schematic of figure 20. The anti-cyclonic pair grows as it is pushed in the direction of rotation, until it completely leaves the field of view at $x/D \approx 0.52$ . The cyclonic primary vortex detaches from the leeward surface and ceases to grow in strength. The counter-flow regions between the cyclonic primary vortex and the surface of the cone take a larger toll on the strength of the vortex; these effects are observed from $\omega _x$ -contours. This further supports the phenomena first observed in the cyclonic primary vortex corresponding to $\theta _c=22.5^\circ$ ; as $\theta _c$ increases, the cyclonic primary vortices experience greater regions of counter-flow that increasingly deteriorate them. The $\varGamma _{{cyc}}$ and $\varGamma _{{anti\hbox{-}cyc}}$ values corresponding to $\theta _c=30^\circ$ , $S_D=2$ and $\alpha = 36^\circ$ are plotted in figure 41. The $\varGamma _{{anti\hbox{-}cyc}}$ values are missing at $x/D = 0.52 {-} 0.72$ as the anti-cyclonic primary vortex is not in the field of view.

Figure 41. $|\varGamma _{{cyc}}|/DU_\infty$ and $|\varGamma _{{anti\hbox{-}cyc}}|/DU_\infty$ versus $x/D$ corresponding to flows over the spinning $\theta _c=30^\circ$ cone at $\alpha =36^\circ$ and $S_D=2$ .

9.1.1. Potential flow streamlines versus streaklines and PIV streamlines

The $\psi$ -contours in red are overlaid streaklines for $S_D=0$ and $\alpha =0$ in figure 44(a). Good agreement with potential flow streamlines is observed near the left edge of the field of view and in regions significantly far from the surface of the cone; however, large disparity is observed with streaklines near the surface as they follow the geometry of the cone and leave the plane of symmetry. Better agreement between the potential flow streamlines overlaid on the corresponding $|\boldsymbol{u_*}|/U_\infty$ contour plot are observed in figure 44(b) due to more precise alignment of the measurement plane with the plane of symmetry.

Figure 42. Smoke streakline images over a cone of half-angle $\theta _c=45^\circ$ , ${\textit{Re}}_D = 3.3 \times 10^4$ , ${\textit{Re}}_L = 2.4 \times 10^4$ , $S_D=0\,\text{and} \pm 1$ , and $0\leqslant \alpha \leqslant 36^\circ$ .

Figure 43. Smoke streakline images over a cone of half-angle $\theta _c=45^\circ$ , ${\textit{Re}}_D = 3.3 \times 10^4$ , ${\textit{Re}}_L = 2.4 \times 10^4$ , $S_D=0\,\text{and}\pm 2$ , and $0\leqslant \alpha \leqslant 36^\circ$ .

Figure 44. $\psi (r,\theta )$ streamlines overlaid streaklines and PIV streamlines at $y\approx 0$ corresponding to $\theta _c=45^\circ$ , $S_D=0$ and $\alpha =0$ .

9.1.2. Streakline behaviours near the surface

The streakline images for $\theta _c = 45^\circ$ vary significantly from the flows corresponding to $\theta _c = 10^\circ , 15^\circ , 22.5^\circ \text{ and }30^\circ$ . Streaklines near the rotating surface of the cone, corresponding to columns 2 and 3 of figures 42 and 43, are shaped in the direction of rotation; increased curvatures are observed in the streaklines corresponding to $|S_D|=2$ , as expected. The stagnation point shifts from the vertex of the cone along the windward surface as $\alpha$ increases. As the half-angle of the cone $\theta _c$ approaches $90^\circ$ , its geometry converges to that of a disc, where a similar behaviour of the stagnation point exists (Kuraan & Savaş Reference Kuraan and Savaş2024). Regularly spaced, wavy rolls also mark the streaklines near the leeward surface of the cone for angles of incidence $\alpha \gtrsim \theta _c$ ; they are present only in the $|S_D|\gt 0$ cases, and hence are signatures of centrifugal instability.

9.1.3. Vortex shedding

As the streaklines enter the wake of the cone for the non-spinning cases, they maintain their integrity for a short distance before breaking up into round patches of smoke. These patches are signatures of periodic vortex shedding. They are visible in column 1, rows 2–6 of figure 42 and column 1, rows 2–6 of figure 43. The effect of rotation appears to prolong the distance into the wake region before the streaklines break up into round patches. Figure 45 shows a close-up view of regular shedding vortex signatures in the wake region behind a cone of half-angle $\theta _c=45^\circ$ , $S_D=0$ and $\alpha = 12^\circ$ , where the full field of view streakline image corresponds to column 1, row 3 of figure 43.

Figure 45. Signatures of shedding vortices in the wake region corresponding to $\theta _c=45^\circ$ , $S_D=0$ and $\alpha = 12^\circ$ .

9.1.4. Flow separation and vortical flows

Regions devoid of streaklines form on both the windward and leeward surfaces of the cone in figures 42 and 43. These regions form as streaklines readily leave the plane of symmetry as they approach the surface of the cone. Surface-attached vortex systems near the leeward surface leave faint helical signatures at angles of incidence $\alpha \gtrsim 30^\circ$ ; hence, $\alpha _*\approx 30^\circ$ and $\varLambda \approx 0.67$ . Figure 46 provides close-up portions near the leeward surface of the cone corresponding to $\alpha =36^\circ$ , and $S_D=0\,\text{and}\pm 2$ (row 7 of figure 43). Figure 46(a) shows faint helical patterns at the leeward surface of the cone, where $S_D=0$ . Signatures of the cyclonic primary vortex are present in the $S_D\gt 0$ and $\alpha = 36^\circ$ cases that are marked by small-scale, wavy rolls; figure 46(b) shows a close-up view of these signatures corresponding to $S_D=2$ . The cyclonic primary vortex remains in close proximity and is aligned parallel to the leeward surface, as the angle of incidence is not high enough for the cyclonic primary vortex to detach. Figure 46(c) shows a close-up region of the $S_D=-2$ and $\alpha =36^\circ$ case, where little to no signs of the anti-cyclonic vortex system are detected. Again, wave-like patterns mark streaklines near the leeward surface that are likely signatures of the boundary layer instabilities revealed by the same spaced, wavy rolls of the $S_D\gt 0$ counterpart.

Figure 46. Close-up sections focused on streaklines near the leeward surface corresponding to $\theta _c=45^\circ$ , $S_D=0\text{ and}\pm 2$ , and $\alpha = 36^\circ$ . (a) $S_D=0$ , (b) $S_D=2$ , (c) $S_D=-2$ .